Introductory Photometry

Transcription

1 Vesion 2.1 Noman Wittels 121 Clak oad Bookline MA Noman Wittels

2 Light This note intoduces the concepts and methods used to measue light. adiant tansfe of enegy happens wheneve photons tavel between objects. When the photons can be seen by humans the adiant enegy is called light. A nice definition of light is visually evaluated adiant enegy 1 because the measuement methods fo light wee developed to match human peception. The basic concepts suounding light pedate much moden technology, including electicity. Theefoe, in the field of photomety (measuement of light), many of the tems 2 come fom the ea when light could only be measued by the human eye. Some vestiges of old usage that still pesist will be mentioned late. 3 The study of light o adiant enegy has been a majo focus 4 of physics eseach fo the past seveal hunded yeas. No attempt will be made to summaize this entie body of wok. Only the pats that ae diectly applicable to machine vision will be discussed. Light can be modelled as photons o as waves. The distinction is most significant when diffaction effects ae involved (fo example, when imaging objects that ae small compaed to the wavelength of light), o when the quantity of light is vey low (because photon statistics become impotant) o vey high (because multiple-photon effects in mateials become impotant). None of these cases descibes nomal uses of light to illuminate objects fo vision - which coesponds to most applications in image pocessing. Theefoe we will use a geometic model that consides light to be made of continuous ays with no diffaction effects. In vacuum light tavels at a constant speed, 3x18m/sec. Accoding to ou cuent theoies of physics, this is the fastest that anything can tavel. Inside mateials light slows down. The atio 1 JF Snell, adiomety and Photomety, in WG Discoll, Handbook of Optics (1978, McGaw-Hill Book Company, New Yok), p An excellent dictionay of lighting teminology can be found in Section 1 of JE Kaufman, ed. IES Lighting Handbook, vol 1 (1981, Illuminating Engineeing Society of Noth Ameica, New Yok). The definitive dictionay is the Intenational Lighting Vocabulay published by the Commission Intenationale de l Éclaiage (CIE-1.1.). 3 The convesion fom peceptual (visual) measuement of light to technical measuement did not pogess diectly to electonic measuement. Photochemical measuement using photogaphic mateials was the fist step. The pimay battles between people who favoed peception vesus those who favoed technology happened about one hunded yeas ago. One account of this stuggle is contained in W.B. Feguson, ed., The Photogaphic eseaches of Fedinand Hute and Veo C Diffield (1974, Mogan and Mogan, Dobbs Fey NY). 4 The eade is asked to begin noticing how visual tems pevade ou language. In his book The Oigins of Knowledge and Imagination (1978, Yale Univesity Pess, New Haven), J Bonowski claims this is because much human knowledge is vision-based. The wods imagine, imaginay, visionay, etc. show this connection clealy. As a esult, humans have difficulty in objectively evaluating light and images. Leaning to conside one's own vision system as an instument with technical stengths and weaknesses which must be studied and undestood befoe it is used as a measuement tool is an impotant step towad success in image pocessing Noman Wittels Vesion 2.1 page 1

3 of the vacuum speed to the intenal speed is called the index of efaction. It is always geate than one. Indices of efaction fo a few mateials ae given in Table 1. Notice that speed of light is almost the same in ai as in vacuum. Table 1. efactive Index fo some Mateials glasses tanspaent plastics 1.5 wate 1.3 ai 1.3 Light tavels in staight lines, called ays, in egions of constant efactive index. When a light ay stikes a bounday sepaating egions with diffeent indices of efaction, two thing happen. Fist, pat of the light is eflected 5. The amount of eflection, which depends on polaization of the light, absoption within the mateials, and diection of incidence, is on the ode of: n = 1 (1) n + 1 whee n=n 2 /n 1 is the atio of the two indices of efaction with n 1 and n 2 selected so that n 1. Fo typical glasses, the faction eflected is on the ode of 4% to 1%, fo the wost combinations of angle and polaization. 1 n 1 n 2 > n 1 n 2 2 Figue 1. Light ay deflection at a change in I. 5 A compendium of infomation on the inteactions between light and mateials can be found in AK Stenius, Nomenclatue and Definitions Applicable to adiometic and Photometic Chaacteistics of Matte, ASTM Special Technical Publication #475, Ameican Society fo Testing and Mateials, Noman Wittels Vesion 2.1 page 2

4 The second thing that happens at boundaies is that the light ay is deflected. Its diection of tavel obeys Snell's law: n sin = n sin whee 1 and 2 ae, espectively, the angles the light ay makes with espect to the nomal on each side of the bounday, Figue 1. When the angles ae small, sin, and tan ae all about equal, which leads to a small-angle appoximation of Equation (2): n n (2) (3) α = 1 n=1 (ai) 1 n>1 Figue 2. Light ay stiking a glass sphee. At cuved boundaies Snell's law is applied locally wheeve a ay stikes the bounday. Conside a glass sphee with adius. A ay stikes the glass at a distance away fom the axis of the sphee, Figue 2. Using the small angle appoximation,.the angle of incidence is = /. Fom equation (2), the efacted angle is 1 = / n. The ay enteed the glass with slope '= and inside the glass it has slope: = = ( )= n 1 tan 1 α α 1 n An object which causes light ays to change slope in popotion to the adius at which they stike it is called a lens. Cuved eflectos ae also lenses. The small angle appoximation avoided having to deal with the fact that a sphee is not a pefect lens (it has a geometical abeation: spheical abeation). Futhe discussion of lenses is beyond the scope of these notes, but many texts books ae available on the subject 6. 6 See, fo example, WHA Fincham and MH Feeman, Optics, 9th edition (198, Buttewoths, London). The Noman Wittels Vesion 2.1 page 3

5 2. Measuement of light Histoically, light was measued by visual compaison. Unknown souces wee compaed with standad light souces by juxtaposition in space (neighboing patches wee compaed to see if thei intensities could be distinguished) o in time (pat of a visual field was apidly switched between souces to see if the flicke wee visible). Photomety is the at of making visual compaisons of light. It is histoically distinct fom adiomety, which is the measuement of adiant enegy in tems of standad SI units 7. The connection between photomety and adiomety oiginally involved standad souces - physical atifacts wee constucted and maintained in standads laboatoies thoughout the wold. Thei light output defined photometic units 8. In the past ten yeas the measuement systems have been combined: photometic units ae now defined in tems of adiometic units. Photomety cuently means adiomety with sensos whose spectal sensitivity match the esponse of the human eye, which vaies by seveal odes of magnitude ove the 38nm to 78nm wavelength ange of visible light. Of the two measuement systems fo adiant enegy, why do we use the olde one? Why ae these notes called Intoductoy Photomety instead of Intoductoy adiomety? At fist glance adiomety seems to make moe sense. Its methods and units have always been connected with moden physics and thee ae no ancient units o concepts to explain. In situations involving monochomatic enegy, such as lase illumination, o invisible adiant enegy, adiomety is pefeable. Howeve, when the illumination is boadband and visible photomety is used fo seveal easons. Fist, most envionmental souces of light (the sun, incandescent and fluoescent lamps, campfies, television sceens, squashed fieflies, otting wood, lightning spheical glass lens is an example in chapte 5. 7 The taditional instument fo measuing adiant enegy is the bolomete, a kind of caloimete in which the tempeatue ise in an iadiated taget of known heat capacity measues adiant enegy tansfe. 8 The oiginal atifacts wee candles that buned contolled amounts of spem whale oil. The photometic unit of luminous intensity is still called the candela; fomely it was called the candle powe. Because of disageements on how to constuct oil candles, diffeent counties had diffeent definitions of the candela. Fom 199 to 1948 banks of vacuum lamps with cabon filaments opeating at contolled cuents wee used as photometic standads. Between 1948 and 1979 the candela was defined as 1/6 the luminous intensity of 1 squae millimetes of glowing ceamic at the melting tempeatue of platinum. Instumentation eos in this definition of the candela wee seveal pecent. In 1979 the photometic and adiometic standads wee meged and luminous intensity is now specified using standad SI units, the mete and the Watt Noman Wittels Vesion 2.1 page 4

6 flashes, boken lifesave candies, etc.) emit boadband adiant enegy. It is easie to discuss them using photomety, which is inheently boadband (emembe, photomety was developed with the human eye as the senso). Second, although cameas do not see as humans, cameas ae used to obseve things that humans find inteesting. Photomety povides simple answes to typical human questions like Which fabic sample is moe eflective, the ed one o the blue one?. That question is difficult to answe using adiometic measuements. Thid, most boadband detectos of visible adiant enegy ae designed to mimic the human visual esponse so photometic measuements apply diectly to them. Fo example, television cameas ae designed with the same light esponse as humans so that television images appea to be natual. Photometic measuements show diectly how an object will appea to a camea. Fo these easons, photomety is useful in applied machine vision. In some cases (eg, monochomatic o invisible adiation, themal adiation) adiomety may be appopiate so equivalent adiometic concepts and units will be mentioned thoughout these notes. 3. Luminous Flux The basic quantity of light, the lumen (abbeviated lm), is a luminous flux. It coesponds to a flux of light which in pinciple epesents some numbe of photons pe second; the adiometic analog is adiant flux, whose units ae W. The actual numbe of photons depends on the spectal content of the light, loosely called colo. A Watt of 55nm (geen) monochomatic light poduces oughly twice the visual sensation of a watt of 61nm (oange) light: its spectal luminous efficiency is twice as lage 9. This function was expeimentally detemined by aveaging the esponses of a lage numbe of humans. It foms the basis of the convesion between adiomety and photomety. To find the lumen content of some flux of light in pinciple one multiplies its spectal enegy distibution (Watts pe nm of wavelength evaluated as a function of wavelength) by the spectal luminous efficiency, which is a tabulated function, and integates ove the ange 38 to 78nm. In fact, the total flux is aely measued this way - it is too difficult and can be highly inaccuate. 1 Flux is measued by putting the unknown souce and a standad souce inside an integating sphee 11, a lage hollow sphee painted matte white 9 The spectal luminous efficiency peaks at 555nm whee its value is 683 lm/w. Its aveage value ove the ange 38 to 78nm is 187 lm/w. 1 A epot of eos geate than 2% when this method is applied to fluoescent lamps (which contain both boadband and spectal line adiation) is contained in B Steine, The Pesent State of adiomety and Photomety, NBS Technical Note 594-6, 1974). 11 EB osa and AH Taylo, Theoy, Constuction and Use of the Photometic Integating Sphee, Scientific Noman Wittels Vesion 2.1 page 5

7 inside. Multiple eflections cause the light to spead out evenly along the sphee's inne suface, even if the souce does not emit unifomly. An electonic senso with filtes to match its sensitivity to the spectal luminous efficiency function measues the light passing though a small hole in the sphee. This flux is popotional to the total flux fom the souce. The flux fom the unknown souce is measued by compaing it with the flux fom the standad souce. At the same time, the luminous efficiency of the souce is obtained by dividing the light output by the electical powe input; its units ae lm/w. 4. Luminous Intensity The luminous intensity of a light souce is the deivative of its flux with espect to solid angle: d I = Φ (4) dω The unit of intensity is the candela (abbeviated cd) - lm/s. The total flux though a point is intensity integated ove all solid angles at that point. The adiometic analog is adiant intensity, whose units ae W/s. Thee ae two ways to measue the intensity of a point souce. Fist, one can measue the flux Φ fom the souce which cosses a pojected aea A at a distance The intensity is just I =Φ 2 / A. This is the same as measuing the flux into a solid angle and dividing by the solid angle. Second, one can use an integating sphee to measue the total souce flux and divide by the total solid angle: I =Φ/4π. This second method aveages the intensity and esults in the mean luminous intensity. A souce which has the same intensity in all diections is isotopic. A small isotopic souce is called a point souce. It is small if its size is much less than the distance to the neaest object it illuminates. The intensity of a point souce in any diection is the same as its mean intensity. A standad 75W incandescent light bulb poduces 117 lm when opeated at 12VAC. Its luminous efficiency is 117/75=15.6 lm/w and its mean luminous intensity is 117/4π=93cd. The actual intensity is lowe than the mean beneath the bulb whee the scew-in connecto blocks the light, so it must be a little highe above the bulb. Papes of the Bueau of Standads, #447, 18, , Noman Wittels Vesion 2.1 page 6

8 Some light souces ae moe anisotopic than the 75W bulb. Automobile headlamps ae designed to put all of the flux into a naow beam. The intensity of this type of souce is specified in beam candelas. Eithe the peak candelas ae specified within a given cone o a diagam of intensity in beam candelas vesus angle is povided. An aicaft landing lamp ated at 4, beam candelas poduces a total flux of 1 lm. Theefoe it emits into a solid angle of 1/ 2 =( ) = = Ω= Φ/ I beam =. 3s. This coesponds to a cone half-angle of α Ω/ π Figue 3 contains a diagam showing convesion between cone angle and solid angle. Aicaft landing light alignment is clealy vey citical. A = π 2 π( α ) 2 α Ω= A 2 =πα2 Figue 3. Steadian equivalent of a small cone angle. 5. Luminance Most light souces ae not infinitesimal point souces, they ae extended souces o finite souces. That is because light flux is elated to powe and natue seems to place maximum limits on powe densities. Extended souces ae descibed by the second deivative of flux with espect to aea and solid angle: L = d dadω 2 Φ (5) The units of luminance ae cd/m 2. The total flux though a suface is the double integal of the luminance ove the suface aea and ove all solid angles at each point in the aea. The adiometic analog is adiance, whose units ae W/(m 2 s). Many othe units of luminance pesist in the liteatue, although thei use is not cuently Noman Wittels Vesion 2.1 page 7

9 fashionable. Some of them ae the nit (nt) - also cd/m 2, the stilb (sb) - cd/cm 2, the apostilb (asb) - cd/πm 2, the Lambet (L) - cd/πcm 2, and the footlambet (fl) - cd/πft 2. a a A Acos Figue 4. Test suface fo studying luminance. Luminance is a diectional quantity; it is necessay to define the diection in which the light is emitted o measued. Conside a plana suface of aea A whose luminance is some function of the pola angle with espect to the suface nomal, L = L( ). It is not assumed that L L( ) is constant, meely that it is independent of the azimuthal angle. This descibes most sufaces obseved in machine vision applications. The total flux though the test aea, a, located a distance away is Φ a = L( ) aa/ 2, see the left side of Figue 4. Note that the test suface nomal has been placed along the adial line connecting the two sufaces. Viewed fom a point on the test suface the luminous suface appeas to have aea A and to be tilted at an angle, its pojected aea is Acos. We can eplace the suface A by one with aea Acos whose nomal points diectly to the test suface, see the ight side of Figue 4. The two cases ae completely equivalent; thee is no measuement that can be made fom the point on the test suface that distinguishes between them. The flux though the test aea in the second case is Φ a = L ( ) aa/ 2, whee L ( ) is the luminance of the new suface. We can equate the two fluxes to find L ( ) = L( )/cos. Thus, if the oiginal suface is otated, the appaent luminance vaies. We can deduce the angula dependence of the oiginal suface's luminance L( ) by measuing the appaent luminance L ( ) and multiplying by the cosine of the otation angle. This is a good point to stop and look aound. Look at cuved objects that ae eithe selfluminous o that have matte finishes: the sun, an egg, the moon, a light bulb, you am, etc. All of these objects appea unifom all of the way to thei edges. That means the appaent luminance Noman Wittels Vesion 2.1 page 8

10 L ( ) = L( )/cos is constant with angle, which can only happen if an object's luminance vaies as the cosine of the obsevation angle. If L( ) fell off moe slowly than cos, the vey edge would be infinitely bight. If L( ) fell off moe apidly than cos, the edge would be totally black. The geneal absence of limb bightening o limb dakening is a demonstation of Lambet's law, which says that an object's luminance is popotional to the cosine of the obsevation angle: L = L cos. (6) Sufaces that obey this law ae called diffuse. Most sufaces that ae not shiny ae diffuse. Fom the analysis suounding Figue 4, the intensity of a luminous souce with aea A that is fa enough away to ignoe the changes in and when moving fom one edge of the souce to anothe is: I = L Acos (7) Notice that the intensity is not isotopic. Viewed on edge, the souce intensity dops to zeo because the pojected aea falls off as cos. This is not a consequence of Lambet's law; it is simple geometic pojection. 6. Illuminance The illuminance at a suface is the deivative of the flux with espect to suface aea: E = dφ da (8) The units of illuminance ae lux (abbeviated lx) - lm/m 2. The total flux though a suface is the integal of illuminance ove its entie aea. Unfashionable units fo illuminance include the footcandle (fc) - lm/ft 2. The concept of illuminance does not distinguish between light that leaves o entes a suface, but two tems that appea in the liteatue do: exitance is illuminance leaving a suface (such as a television sceen) and incidance is illuminance enteing a suface (such as the gass on a golf couse). The adiometic analog is iadiance, whose units ae W/m Noman Wittels Vesion 2.1 page 9

11 E = I 2 E = I cos 2 aea A intensity I Figue 5. Illuminance due to a Point Souce. aea A Illuminance is a diectional quantity so it is necessay to specify the diection of the test suface. Fo example, a hoizontal test suface on the eath at noon in the summe in New England eceives an illuminance of about 1 5 lx; a vetical test suface facing south eceives about 4x1 4 lx. The illuminance at a suface facing a point souce with intensity I located a distance away is E = I/ 2. This is the famous invese squae law fo illuminance. If the suface is inclined at an angle ø as shown in Figue 5 the illuminance is: E = I cos 2 (9) E S E = E S cos Figue 6. Illuminance fom a Distant o Collimated souce. The cos is due to geometic pojection, not Lambet's law. Note that illuminance is a diffeential quantity measued at a point. On an extended suface, will in geneal be diffeent fom point to point, so E will also vay acoss the suface. Thee ae cases whee and ae not significantly diffeent fom point to point: a souce that is fa away o a well-collimated (low divegence angle) souce, such as a lase beam. In those cases it is appopiate to model the Noman Wittels Vesion 2.1 page 1

12 souce as poviding an illuminance E S and to calculate the illuminance it poduces on a taget using E = E S cos, Figue 6. Calculate the illuminance at a point on a diffuse suface with axial luminance L. This is just the integal of the luminance ove all solid angles in a hemisphee: 2π E = L ( Ω) d Ω (1) Use the fact that dω= da/ 2, whee da is the diffeential aea on the suface of a hemisphee with adius. This integal is easiest to pefom in pola coodinates, see Figue 7: 2 da L E = L cos = d d = L A ϕ cossin π 2 2 2π π / 2 (11) z ϕ sin sindϕ sin dϕ d y x d da = 2 sin dϕ d Figue 7. Illuminance of a Diffuse Object. The Illuminance of a diffuse object is π times its luminance 12. The 75W incandescent lamp with 117 lm flux can be modelled as a sphee 6mm in diamete. Its suface aea is A=π(.6) 2 =.11m 2 so its suface illuminance is 117/.11=13klx and its suface luminance is 13,/π=33kcd/m 2. Note this elationship is only tue fo diffuse objects: fosted lamps ae highly diffuse. In the geneal case Equation (1) is used to elate luminance to illuminance. 12 This is the oigin of the pi in luminance units such as the Lambet. It makes the luminance and illuminance of a diffuse object numeically equal Noman Wittels Vesion 2.1 page 11

13 The Illuminance poduced by an extended souce at a point on a test suface is found by calculating the diffeential illuminance caused by a small patch at a point P on the souce. Assume the patch is a diffuse emitte with luminance L, aea da, and it is located a distance away fom the test suface point, Figue 8. Then the diffeential illuminance is: de L 2 cos1cos2 = da 2 (12) This is just the combination of Equations (7) and (9). Using a diffeential aea allowed us to model the light fom the point P as being an intensity. To find the total luminance at the test point, integate equation (12) ove the entie suface aea of the souce: L E = L A cos cos da (13) Test Suface 2 1 P Souce Figue 8. Illuminance due to an Extended Souce. Souce point P aea da h Test Suface Figue 9. Illuminance when Souce is Paallel to Test Suface Noman Wittels Vesion 2.1 page 12

14 As an example conside the illumination due to a diffuse plana souce a distance h away fom and paallel to the test suface, Figue 9. The adial distance fom P on the souce to the test suface is = h/ cos. Fom Equation (13), the illuminance is: 4 E = L cos da A (14) Disk Souce adius ϕ dρ ρ ρ h ρ = h tan dϕ h dρ = cos 2 d Test Suface Figue 1. Illuminance when the Souce is a Disk. da = ρ dρ dϕ If the souce is a disk of adius, Figue 1, the illuminance is: 2π 1 tan E L 2 H 4 2πLh = d d = d = L 2 h ϕ ρcos ρ 2 h sincos π 2 + h 2 2 Note that in the limit as, the illuminance educes to E =πl, which is just the illuminance of the souce. Objects facing lage plana souces eceive the same illuminance as the souce emits. The method used fo calculating the illuminance caused by a disk can be extended to souces with othe geometies eflections So fa, we have consideed only self-luminous objects. Most objects ae luminous because they eflect light. The tem eflection coves a lage ange of phenomena. We ae pimaily inteested in two cases: specula eflection and diffuse eflection. Specula eflection is mio- 13 Calculations fo a numbe of souce geometies ae contained in Z Yamanouti, Geometical Calculations of Illumination due to Light fom Luminous Souces of Simple Foms, es. Electotech. Lab. (Jpn), 148, 1924, and Futhe Study of Geometical Calculation of Illumination due to light fom Luminous Suface Souces of Simple Foms, ibid., 194, Noman Wittels Vesion 2.1 page 13

15 like eflection in which a light ay eflects fom the suface without diffusion. The incident and eflected ays lie in the same plane as the nomal to the suface and they fom equal angles with it at the point of incidence. Diffuse eflection is the case in which the eflected light is independent of illumination diection. No mateials fit eithe of these cases exactly although often many mateials can be adequately modelled by a combination of specula and diffuse eflections. 14 Othe desciptions of eflection, which will be excluded fom these notes, include: bloom - eflection nea the specula angle due to deposits on the suface, haze - which is like bloom except that the scatteing mateials ae pat of the suface and cannot be emoved, sheen - specula eflection at lage angles of incidence fo an othewise matte specimen, and specula eflection distinctness - the shapness with which outlines ae eflected by the suface. Also, the eflection models used in compute gaphics to make images that look good (eg., models which educe the attention the eye places on idges whee the plana patches that fom thee-dimensional images meet) have been omitted 15. Such models ae beyond the scope of these notes. L L = L s s Object Appaent Object Suface Nomal ay fom Object S Mio Figue 11. Specula eflection. Specula eflection is chaacteized by a specula eflectivity s, which is the atio of eflected flux to incident flux. The eflecting suface, is not seen. All ays that appea to come fom the suface oiginate at some othe suface whose location can be detemined by using the equality of the angles of incidence and eflection, Figue 11. If the object has luminance L, then 14 N Wittels and SH Zisk, Lighting Design fo Industial Machine Vision, Poc. SPIE, 728, 47-56, Two standad efeences ae the papes ae by BT Phong, Illumination fo Compute Geneated Pictues, Comm. ACM, 18, , 1975 and H Gouaud, Continuous Shading of Cuved Sufaces, IEEE Tans. Computes, C-2, , Noman Wittels Vesion 2.1 page 14

16 it appeas as if the object has luminance L s. The impotant distinction about specula eflection is that incident ays etain thei diectional infomation duing eflection. Diffuse eflection is chaacteized by a diffuse eflectivity d, which is the atio of eflected flux to incident flux. With diffuse eflection thee is a total loss of infomation about the diection of incidence. The two cases shown in Figue 12 ae totally equivalent - no measuement made on the eflected light can distinguish between them. Whethe the illumination comes fom a point souce, an extended souce, o any othe type of souce, the amount of light diffusely eflected depends only on the total illumination of the eflecto. If the incident illuminance is E ( E = Icos / 2 fo the cases shown), then the eflected illuminance is E d. Since the suface is diffuse the axial luminance of the eflected light is L = E d /π; the luminance in othe diections is descibed by equation (6). Intensity I Extended Diffuse Souce d Diffuse eflecto Figue 12. Diffuse eflection. d L = I cos π 2 Specula and diffuse eflection can be combined to descibe the total eflection 16 which is chaacteized by its total eflectivity t, the eflected flux divided by the incident flux: t = s + d The total eflectivity t must always be less than one. Its components ae measued in two expeiments. Fist, diffuse eflectivity is measued by illuminating with a known flux along a known diection and measuing the total eflected flux in all diections except the specula one, see the left side of Figue 13. The hole in the detecto is impotant: it excludes speculaly eflected light. Hemispheical detectos ae difficult to make so the aangement shown on the ight side of Figue 13 can be used. It is fundamental that all systems which tansfe light must 16 This is a simplification that ignoes polaization effects. They can be impotant because specula eflection geneally peseves o enhances linea polaization while diffuse eflection geneally educes o eliminated linea polaization. (16) Noman Wittels Vesion 2.1 page 15

17 be evesible: if souce and detecto ae exchanged, the same measuements will esult. 17 Next the specula eflectivity is measued using equal angles of incidence and measuement, Figue 14. The contibution fom diffuse eflection to the measued flux in this expeiment must be subtacted to get the coect specula eflectivity. Note that Figues 13 and 14 ae schematic diagams only. eflectometes ae moe complex than these diagams, but photometic instument design is beyond the scope of these notes. Souce Detecto Hole Incident Light Hemispheical Detecto Hole eflected Light Hemispheical Souce Test Sample Test Sample Figue 13. Measuing Diffuse eflectivity. Souce Incident Light Detecto Test Sample eflected Light Patially eflecting Mio Figue 14. Measuing Specula eflectivity. 8. Spectal Content Photomety is based on the notion that the sensos match the spectal sensitivity of the human eye. The situation is usually moe complicated. Fist, human spectal sensitivity is diffeent unde low and high flux conditions. Also individual humans diffe in thei sensitivities 17 This is a esult of the second law of themodynamics. An optical system tansfes adiant enegy. If the enegy flowing fom souce to detecto wee diffeent afte the exchange, it would fom the basis fo a pepetual motion machine. Of couse systems which violate evesibility can be built, but they equie enegy input (Maxwell's demon must be fed) Noman Wittels Vesion 2.1 page 16

18 - which is one eason that the spectal luminous efficiency function is now standadized in tems of SI units instead of being standadized by measuements on people. Finally, no camea pefectly matches the human spectal esponse. The solution is to expand all photometic units to include the notion of deivatives with espect to wavelength. They ae denoted by adding the wods "spectal" in font of the tems. Fo example, spectal luminance is the deivative with espect to aea, solid angle, and wavelength: L λ d = dadωdλ 3 Φ (17) It has units cd/m 3, since wavelength has units of length. All of the calculations done in these notes ae valid fo the coesponding spectal units. Any of the photometic units can be ecoveed by integating the espective spectal units ove all wavelengths between 38 and 78nm: 78nm 78nm E = E dλ; L = L dλ, λ sλ λ 38nm 38nm etc. One final wod of caution is in ode egading spectal photometic units. They descibe spectal content of light, which is not the same as colo. Colo is a combined physiological and psychological esponse to light. The spectal content of light detemines in pinciple the colo of the light but the detemination is not unique. A colo desciption does not imply a paticula spectal distibution. That is fotunate because it allow us to poduce colo pictues and photogaphs using mixtues of only thee o fou inks o dyes. Matching spectal content is much moe difficult than matching colo: at least an ode of magnitude incease in the numbe of inks o dies would be equied. An expeimentally deived set of pimay colos (ed, geen, blue) and a coesponding set of thee spectal luminous efficiency functions has been found to give acceptable esults when matching colos. This model is called the ti-chomatic system. Thus, colo calculations can be pefomed with sets of thee spectally weighted analogs fo each of the photometic functions mentioned above. Futhe discussion of colo is beyond the scope of this intoduction but it can be found elsewhee The book by TN Consweet, Visual Peception (197, Academic Pess) has a lengthy discussion of the physiological bases fo colo esponse in people. A desciption of how the system is used is contained in the book by DB Judd and G Wyszecki, Colo in Business, Science, and Industy (1963, J Wiley, NY). A eview aticle on colo has been witten by G Wyszecki, The Measuement of Bightness and Colo, Metologia, 2, , Noman Wittels Vesion 2.1 page 17